Abstract: An explicit expression is given in the frequency domain for the minimum free
energy related to a particular state of a linear rigid heat conductor with memory
effects in the constitutive equations, using the fact that this quantity
coincides with the maximum recoverable work obtainable from that state.
The constitutive equations for the internal energy and the heat
flux are expressed as linear functionals of the histories of temperature
and its gradient, respectively, together with the present value of the
latter quantity. Another equivalent expression for the minimum free energy
is also deduced and used to derive explicit formulae for a discrete spectrum
model.

Abstract: In this paper, we study the solid combustion system
with the monostable nonlinearity $f(T)=T$.
Our goal is to prove the existence of pulsating waves.
First, the specific form of the nonlinearity leads to transform the
problem into
a scalar reaction diffusion equation with nonusual infinite limits.
Next, we prove that this scalar equation admits a family of pulsating waves,
applying an easy fixed point argument;
moreover we precise the asymptotic behaviour of the pulsating waves,
developping them in Fourier series and studying the behaviour
of its Fourier coefficients.
Finally, all these informations on the equation let us prove that there exists
a family of pulsating waves of the
original SHS system, the family of admissible propagation speed being
a precised half-line.

Abstract: This paper focuses on the parametric abundance and the 'Cantorial'
persistence under perturbations of a recently discovered class of strange
attractors for diffeomorphisms, the so-called quasi-periodic
Hénon-like. Such attractors were first detected in the Poincaré map of a
periodically driven model of the atmospheric flow: they were characterised
by marked quasi-periodic intermittency and by
$\Lambda_1>0,\Lambda_2\approx0$, where $\Lambda_1$ and $\Lambda_2$ are the
two largest Lyapunov exponents. It was also conjectured that
these attractors coincide with the closure of the unstable manifold of a
hyperbolic invariant circle of saddle-type.
This type of attractor is here investigated in a model map of the solid
torus, constructed by a skew coupling of the Hénon family of planar maps
with the Arnol$'$d family of circle maps. It is proved that Hénon-like
strange attractors occur in certain parameter domains. Numerical evidence
is produced, suggesting that quasi-periodic circle attractors and
quasi-periodic Hénon-like attractors persist in relatively large subsets
of the parameter space. We also discuss two problems in the numerical
identification of so-called strange nonchaotic attractors and the
persistence of all these classes of attractors under perturbation of the
skew product structure.

Abstract: This paper is devoted to a mathematical and numerical study of a simplified kinetic model for evaporation phenomena in gravitational systems. This is a first step towards a mathematical understanding of more realistic kinetic models in this area. It is well known in the astrophysics literature that the appropriate kinetic model to describe escape (evaporation) from stars clusters
is the so-called Vlasov-Landau-Poisson system with vanishing boundary condition at positive microscopic energies. Since collisions between stars and their self-consistent interactions are both taken into account in this model, its mathematical analysis is difficult, and so far not achieved. Here, as a first step, we focus on a simplified framework
of this model and make the following assumptions: i) Only homogenous (space-independent) distributions functions are considered, leading to a collisional kinetic model with a vanishing boundary condition in velocity. ii) The interaction potential involved in the Landau collision operator is of Maxwellian type. iii) The escape velocity (or energy) is supposed to be constant. Using these assumptions, we first establish the well-posedness of the associated Cauchy problem. Then, we focus on the long time behavior of the solution and prove that the energy of the system decreases in time as $O(1/\log(t))$ (logarithmic evaporation), with convergence to a Dirac distribution in velocity when time goes to infinity. Finally, a suitable numerical scheme is constructed for this model and some simulations are performed
to illustrate the theoretical study.

Abstract: We propose a method for numerical integration of Wasserstein gradient flows
based on the classical minimizing movement scheme.
In each time step, the discrete approximation is obtained as the solution
of a constrained quadratic minimization problem on a finite-dimensional function space.
Our method is applied to the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation,
which arises in quantum semiconductor theory.
We prove well-posedness of the scheme and derive a priori estimates on the discrete solution.
Furthermore, we present numerical results which indicate
second-order convergence and unconditional stability of our scheme.
Finally, we compare these results to those
obtained from different semi- and fully implicit finite difference discretizations.

Abstract: We consider a discontinuous map with square-root singularity, which is
relevant to many physical systems. Such maps occur in modeling
grazing-sliding bifurcations in switching dynamical systems, or if the
Poincaré plane coincides with the switching plane. It is shown that
there are notable differences in the bifurcation scenarios between
this type of discontinuous map and a continuous map with square-root
singularity. We determine the bifurcation structures and the scaling
constant analytically. A different kind of period increment is
observed, and the possibility of breakdown of period increment cascade
is detected. Finally, we show that a system of piecewise smooth
ordinary differential equations can exhibit the same type of
bifurcation behavior.

Abstract: Many times in dynamical systems one wants to understand the bounded motion
around an equilibrium point. From a numerical point of view, we can take
arbitrary initial conditions close to the equilibrium points, integrate the
trajectories and plot them to have a rough idea of motion. If the dimension of
the phase space is high, we can take suitable Poincaré sections and/or
projections to visualise the dynamics. Of course, if the linear behaviour
around the equilibrium point has an unstable direction, this procedure is
useless as the trajectories will escape quickly. We need to get rid, in some
way, of the instability of the system.
Here we focus on equilibrium points whose linear dynamics is a cross product of
one hyperbolic directions and several elliptic ones. We will compute a high
order approximation of the centre manifold around the equilibrium point and use
it to describe the behaviour of the system in an extended neighbourhood of this
point. Our approach is based on the graph transform method. To derive an
efficient algorithm we use recurrent expressions for the expansion of the
non - linear terms on the equations of motion.
Although this method does not require the system to be Hamiltonian, we have
taken a Hamiltonian system as an example. We have compared its efficiency with
a more classical approach for this type of systems, the Lie series method. It
turns out that in this example the graph transform method is more efficient
than the Lie series method. Finally, we have used this high order approximation
of the centre manifold to describe the bounded motion of the system around and
unstable equilibrium point.

Abstract: We consider compressible viscoelastic fluids satisfying the Oldroyd
constitutive law. We prove the local existence (and uniqueness) of
flows by a classical fixed point argument. We also prove some global
properties of the solutions. In particular, we obtain some a priori
estimates which are uniform in the Mach number and prove global
existence of weakly compressible fluids flows. We show that weakly
compressible flows with well-prepared initial data converge
to incompressible ones when the Mach number converges to
zero.

Abstract: We propose a spectral collocation method for solving initial value
problems of first order ODEs, based on
the Legendre-Gauss-Lobatto interpolation. This method is easy to be implemented
and possesses the spectral accuracy. We also develop a multi-step version of this
process,
which is very
available for long-time calculation. Numerical results demonstrate
the high accuracy of suggested algorithms and coincide well with
the theoretical analysis.

Abstract: Mesenchymal motion describes the movement of cells in biological tissues formed
by fibre networks. An important example is the migration of tumour cells through
collagen networks during the process of metastasis formation. We investigate the
mesenchymal motion model proposed by T. Hillen in [14] in higher
dimensions. We formulate the problem as an evolution equation in a Banach space
of measure-valued functions and use methods from semigroup theory to show the
global existence of mild and classical solutions. We investigate steady states of the
model and show that patterns of network type exist as steady states. For the
case of constant fibre distribution, we find an explicit solution and we prove
the convergence to the parabolic limit.

Abstract: In this work, a predator-prey model with predators impulsively
diffusing between two patches is investigated. By the stroboscopic
map of the discrete dynamical system, the prey-extinction periodic
solution of the investigated system is proved to be globally
asymptotically stable. By the theory of impulsive differential
equation, the investigated system is also proved to be permanent.
Finally, the numerical simulation is inserted to illustrate the
results.

Abstract: Identification of conditions for stable coexistence of interacting
populations is a problem of the highest priority in mathematical biology.
This problem is usually considered under specific assumptions made
regarding the functional forms of non-linear feedbacks. Apparently,
such an approach is lacking generality. In this paper, we consider
the dynamics of two species with interaction of predator--prey (consumer-supplier)
type with the assumption that a part of the resource is neglected
or wasted by the predator (consumer). This model describes, for instance,
killing for fun; such behaviour is typical for many predators when
the prey is abundant.
We assume that the functional responses that are usually included
in such models are given by unspecified functions. Using the direct
Lyapunov method, we derive the conditions which ensure global asymptotic
stability of this model. It is remarkable that these conditions impose
much weaker constraints on the system properties than that are usually
assumed.

Abstract: We are concerned with a reaction-diffusion-advection system proposed by
Hildebrand [4].
This system is
a phase transition model arising in
surface chemistry.
For this model,
several stationary patterns have been shown
by the numerical simulations (e.g., [15]).
In the present paper,
we obtain sufficient conditions for the existence
(or nonexistence)
of nonconstant stationary solutions.
Our proof is based on the Leray-Schauder
degree theory.
Some a priori estimates for solutions play an
important role in the proof.

Abstract: For a general nonlinear diffusion-convection equation, the existence
of uncountably infinite many global monotonic wavefront solutions
and semi-wavefront solutions with bounded support is proved. By
using the method of planar dynamical systems, the dynamical behavior
of the corresponding traveling wave system is discussed. For some
concrete nonlinear diffusion-convection equations, more than thirty
exact explicit parametric representations of the wavefront
solutions, semi-wavefront solutions and unbounded traveling wave
solutions are given.

Abstract: An indefinite weight eigenvalue problem characterizing the threshold
condition for extinction of a population based on the single-species
diffusive logistic model in a spatially heterogeneous environment is
analyzed in a bounded two-dimensional domain with no-flux boundary
conditions. In this eigenvalue problem, the spatial heterogeneity of
the environment is reflected in the growth rate function, which is
assumed to be concentrated in $n$ small circular disks, or portions of
small circular disks, that are contained inside the domain. The
constant bulk or background growth rate is assumed to be spatially
uniform. The disks, or patches, represent either strongly favorable or
strongly unfavorable local habitats. For this class of piecewise constant
bang-bang growth rate function, an asymptotic expansion for the
persistence threshold λ1, representing the positive principal
eigenvalue for this indefinite weight eigenvalue problem, is
calculated in the limit of small patch radii by using the method of
matched asymptotic expansions. By analytically optimizing the
coefficient of the leading-order term in the asymptotic expansion of
λ1, general qualitative principles regarding the effect of
habitat fragmentation are derived. In certain degenerate situations,
it is shown that the optimum spatial arrangement of the favorable
habit is determined by a higher-order coefficient in the asymptotic
expansion of the persistence threshold.

Abstract: A rigorous numerical proof for establishing existence of a
transversal homoclinic orbit for a saddle fixed point with higher
dimensional unstable eigenspaces is presented. As the first
component of this method, a shadowing theorem that guarantees the
existence of such a homoclinic orbit near a suitable pseudo orbit
given the invertibility of a certain Jacobian is proved. The
second component consists of a refinement procedure for
numerically computing a pseudo homoclinic orbit with sufficiently
small local errors so as to satisfy the hypothesis of the theorem.
The third component verifies that the homoclinic orbit is
transversal. In [6], they proved the existence of
transversal homoclinic orbits near anti-integrable limits and near
singularities for the Arneodo-Coullet-Tresser maps. In this paper,
the existence of transversal homoclinic orbits were proved far
away from anti-integrable limits and singularities for these maps.

Abstract: In this paper, we investigate a class of doubly degenerate parabolic equations with periodic sources
subject to homogeneous Dirichlet boundary conditions.
By means of the theory of Leray-Schauder degree,
we establish the existence of non-trivial nonnegative periodic solutions.
The key step is how to establish the uniform bound estimate of approximate solutions,
for this purpose we will make use of Moser iteration and some results of the
eigenvalue problem for the $p$-Laplacian equation.

Abstract: This paper concerns a convective nonlinear diffusion equation
which is strongly degenerate. The existence and uniqueness
of the $BV$ solution to the initial-boundary problem are proved.
Then we deal with the anti-shifting phenomenon
by investigating the corresponding
free boundary problem. As a consequence, it is possible to find a
suitable convection such that the discontinuous point of the
solution remains unmoved.

Abstract: In this paper we consider the persistence of lower dimensional
elliptic invariant tori with prescribed frequencies in reversible
systems, and prove that if the frequency mapping has non-zero Brouwer's
degree at a certain point that satisfies Melnikov's non-resonance
conditions, then the invariant torus with given
frequency persists under small perturbations.

Abstract: The dynamics of magnetization under the applied spin current is
modeled by the generalized Landau-Lifshitz-Gilbert
equation with a spin transfer torque term. Using matched
asymptotic expansion with the domain wall thickness $\epsilon$ as
the small parameter, we derive analytically the dynamic law for
the domain wall motion induced by the spin current. We show that
the domain wall driven by adiabatic current spin-transfer torque
moves with a decreasing velocity and eventually stops. With a
pinning potential, the domain wall motion is a damped oscillation
around the pinning site with an intrinsic frequency which is
independent of the strength of the current. When the AC current is
applied, the dynamic law shows that the frequency of the applied
current can be turned to maximize the amplitude of the
oscillation. The results obtained are consistent with the recent
experimental and numerical results.